To translate statements stated in English using a given set of
predicate symbols, we first restate English proposition using the predicates, connectives, and
quantifiers such that it preserve its original meaning. Then replace the English phrases with the corresponding symbols
.
Example 1: Given the sentence "Not every integer is even".
Now let the predicate "E(x)" represent x is even, and that the universe is the set of integers, First restate it as "It is not the case that every integer is even"
Then "it is not the case" can be represented by the connective "
",
"every object x in the universe" by "
x", and "x
is even" by E(x).
Thus altogether wff becomes
x E(x).
.
Example 1: Given the sentence "Not every integer is even".
Now let the predicate "E(x)" represent x is even, and that the universe is the set of integers, First restate it as "It is not the case that every integer is even"
Then "it is not the case" can be represented by the connective "
",
"every object x in the universe" by " x", and "x
is even" by E(x).
Thus altogether wff becomes
x E(x).
Example 2
Take universe of discourse a set of all students of Kathmandu College.
P(x)
represents: x takes
Discrete Mathematics class.
Here universal quantification is ∀x P(x), which represent the English sentence “all
students of Kathmandu college take Discrete Mathematics class”, and now it is a proposition.
The universal quantification is conjunction of all the propositions
that are obtained by assigning the value of the variable in the predicate.
Going back to above example if universe of discourse is a set {Ram, Shyam, Hari,
Sita} then the truth value of the universal quantification is given by P(ram) ∧ P(Shyam) ∧ P(Hari) ∧ P(Sita) i.e. it is true only if all the atomic propositions are true.
Existential Quantifier
Universal quantifier, denoted by ∃, is used for existential quantification. The
existential quantification of P(x), denoted by ∃x P(x), is a proposition “P(x)
is true for some values of x in the universe of discourse”. The other forms of
representation include “there exists x such that P(x) is true” or “P(x) is true
for at least one x”.
Example 3
For the same
problem given in
universal quantification ∃x P(x)
is a proposition
is represent like “ some students
of Kathmandu College take Mathematics class”.
The existential quantification is the disjunction of all the
propositions that are obtained by assigning the values of the variable from the
universe of discourse. So the above example is equivalent to P(Ram) ∨
P(Shyam)∨P(Hari)∨ P(Sita), where all the
instances of variable are as in example of
universal quantification. Here if at least one of the students takes graphics
class then the existential quantification results true.
Translating the
Sentences into Logical Expression
Example 4
Translate “not every integer is even” where the universe of discourse
is set of integers.
Solution
Let E(x) denotes x is even.
Then ¬∀xE(x) represents the above statement “not every integer
is even”
Example 5
Translate “every man is mortal”
Let M(x) denote x is mortal, where x is from set of man (here universe
of discourse is all man)
Then, ∀x M(x)
represent that “for all x , x is mortal.”
